Tutte polynomial of pseudofractal scale-free web
arXiv:1207.6864v2 [math-ph] 25 May 2013
Tutte polynomial of pseudofractal scale-free web Junhao Peng a,b,c Guoai Xu b a School
of Mathematics and Information Science, Guangzhou University , Guangzhou 510006 , China.
b State
Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications , Beijing 100876 ,China.
c Key
Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University,Guangzhou 510006 ,China.
Abstract The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both combinatorics and statistical physics. It contains various numerical invariants and polynomial invariants ,such as the number of spanning trees,the number of spanning forests , the number of acyclic orientations , the reliability polynomial,chromatic polynomial and flow polynomial . In this paper,we study and gain recursive formulas for the Tutte polynomial of pseudofractal scale-free web(PSW) which implies logarithmic complexity algorithm is obtained to calculate the Tutte polynomial of PSW although it is NP-hard for general graph.We also obtain the rigorous solution for the the number of spanning trees of PSW by solving the recurrence relations derived from Tutte polynomial ,which give an alternative approach for explicitly determining the number of spanning trees of PSW.Further more,we analysis the all-terminal reliability of PSW and compare the results with that of Sierpinski gasket which has the same number of nodes and edges with PSW. In contrast with the well-known conclusion that scale-free networks are more robust against removal of nodes than homogeneous networks (e.g., exponential networks and regular networks).Our results show that Sierpinski gasket (which is a regular network) are more robust against random edge failures than PSW (which is a scalefree network) .Whether it is true for any regular networks and scale-free networks ,is still a unresolved problem. Key words: Pseudofractal scale-free web, Tutte polynomial, Spanning trees, Reliability polynomial PACS: 05.45.Df, 89.75.Hc, 05.10.-a
Email address: [email protected] (Junhao Peng).
Preprint submitted to Physica A
2 May 2014
1
Introduction
The Tutte polynomial is a two-variable polynomial which can be associated with a graph, a matrix, or, more generally, with a matroid . This polynomial was introduced by W.T. Tutte [1–3] and has many interesting applications in several areas of sciences such as combinatorics, probability, statistical mechanics and computer science. It is quite interesting since several combinatorial, enumerative and algebraic properties of the graph can be investigated by considering special evaluations of it [4]. For instance, one gets information about the number of spanning trees [5, 6], spanning connected subgraphs [7], spanning forests [8] and acyclic orientations [9] of the graph by evaluating tutte polynomial at particular points (x, y) . Moreover, the Tutte polynomial contains several other polynomial invariants, such as flow polynomial [10] , reliability polynomial [11] and chromatic polynomial [2, 12, 13]. It has also many interesting connections with statistical mechanical models such as the Potts model [14–16] and the percolation [17]. Despite its ubiquity,there are no widely-available effective computational tools to compute the Tutte polynomials of a general graph of reasonable size.It is shown that, many of the relevant coefficients do not even have good randomised approximation schemes and various decision problems based on the coefficients are NP-hard [18,19].Although it is hard to to compute the Tutte polynomials,a lot of efforts have been devoted to the study the Tutte polynomials of different graph such as polygon chain graphs [16], Sierpinski gaskets [20] and strips of lattices [21–24].However, polygon chain ,lattices and Sierpinski gaskets cannot well mimic the real-life networks, which have been recently found to synchronously exhibit two striking properties: scale-free behavior [25] and small-world effects [26]. Pseudofractal scale-free web(PSW) we studied is a deterministically growing network introduced by S.N. Dorogovtsev [27] which is used to model scale-free network with small-world effect.Lots of job was devoted to study its properties ,such as degree distribution ,degree correlation , clustering coefficient [27, 28] ,diameter [28],average path length [29], the number of spanning trees [30] and mean first-passage time for random walk [31].As for its tutte polynomial and reliability polynomial ,to the best of our knowledge, related research was rarely reported . In this paper,we study and gain recursive formulas for the Tutte polynomial of PSW .The analytic method is based on the its recursive construction and self-similar structure .Recursive formulas for various invariants of Tutte polynomial can also obtained based on their connections with the Tutte polynomial . We also obtain the rigorous solution for the the number of spanning trees by solving the recurrence relations derived from the Tutte polynomial,which coincides with the result obtained in [30] .Thus we give an alternative approach for explicitly determining the number of spanning trees of PSW .Further more,we 2
analysis the all-terminal reliability of PSW and compare the results with that of Sierpinski gasket which has the same number of nodes and edges with PSW. In contrast with the well-known conclusion that scale-free networks are more robust against random node failures than homogeneous networks (e.g., exponential networks and regular networks) .Our results show that Sierpinski gasket (which is a regular network) are more robust against random edge failures than PSW (which is a scale-free network) .
2
Preliminaries
2.1 The Tutte polynomial Let G = (V (G), E(G)) denotes a graph with vertex set V (G) and edge set E(G); we will often write V and E, when there is no risk of confusion, and so G = (V, E). A subgraph H = (V (H), E(H)) of a graph G = (V (G), E(G)) is said spanning if the condition V (H) = V (G) is satisfied. In particular, a spanning tree of G is a spanning subgraph of G which is a tree.Let H be a spanning subgraph of G and k(H) be the number of connected components of H.then the rank r(H) and the nullity n(H) of H are defined as r(H) = |V (H)| − k(H) = |V (H)| − k(H) n(H) = |E(H)| − r(H) = |E(H)| − |V (H)| + k(H)
Definition 1 Let G = (V, E) be a graph. The Tutte polynomial T (G; x, y) of G is defined as [4] T (G; x, y) =
X
(x − 1)r(G)−r(H) (y − 1)n(H)
(1)
H⊆G
where the sum runs over all the spanning subgraphs H of G . The Tutte polynomial can be evaluated at particular points (x, y) to give numerical invariants,such as the number of spanning trees, the number of forests and the number of connected spanning subgraphs.The following theorem [4] depicts the relations between the Tutte polynomial and its numerical invariants. Theorem 1 Let G = (V, E) be a connected graph ,Then: (1) T (G; 1, 1) is the number of spanning trees; (2) T (G; 1, 2) is the number of spanning connected subgraphs of G; 3
Fig. 1. Growth process for PSW from n=0 to n=2
(3) T (G; 2, 1) is the number of spanning forests of G; The Tutte polynomial also has a variety of single-variable polynomial invariants associated with the graph,such as the all-terminal reliability polynomial,flow polynomial and the chromatic polynomial.In this paper we only study the all-terminal reliability polynomial. Let G = (V, E) be a graph,each edge of G has a known probability p of being operational; otherwise it is failed. Operations of different edges are statistically independent,while the nodes of G never fail. The all-terminal reliability polynomial R(G, p) of G is defined as is the probability that there is a path of operational edges between any pair of vertices of G.The connection between the Tutte polynomial and all-terminal reliability polynomial is given by the following theorem [4]. Theorem 2 Let G = (V, E) be a graph, Then R(G, p) = p
|V (G)|−1
(1 − p)
|E(G)|−|V (G)|+1
1 T G; 1, 1−p
!
(2)
2.2 Structure of PSW The scale-free network we studied is a deterministically growing network which can be constructed iteratively [27] . We denote the pseudofractal scale-free web(PSW) after n iterations by G(n) with n ≥ 0. Then it is constructed as follows: For n = 0, G0 is a triangle. For n ≥ 1, Gn is obtained from Gn−1 : every existing edge in Gn−1 introduces a new node connected to both ends of the edge. The construction process of the first three generation is shown in Fig. 1. The PSW exhibits some typical properties of real networks. Its degree distribution P (k) obeys a power law P (k) ∼ k 1+ln 3/ ln 2 [27], the average path length scales logarithmically with network order [29]. The network also has an equivalent construction method [30, 33],as can be seen in Fig. 2 : to obtain G(n + 1), one can make three copies of G(n) and join them at the three most 4
C
B A G1
C
C G2
A B
G1
B A
G2
A
G3
B G3
C
Fig. 2. Second construction method of PSW: G(n + 1) is composed of three copies of G(n) denoted as G1 ,G2 ,G3 ,the three hub nodes of which are represented by A,B,C in the corresponding triangle(The left side of the figure ).In the merging process,hub node A of G1 and hub node B of G3 ,hub node A of G3 and hub node B of G2 , hub node A of G2 and hub node B of G1 are identified as a hub node A,B,C of G(n + 1) respectively(The right side of the figure ).
connected nodes denoted by A,B,C, which are called the hub nodes of G(n+1) in this paper. According to the second construction algorithm, one can see that at each step n, the total number of edges in the network increases by a factor of 3. Thus, the total number of edges for G(n) is En = 3n+1 .We can also find n+1 that the total number of nodes for G(n) is Vn = 3 2 +3 .
3
Tutte polynomial of PSW
Let us simply denote by Tn (x, y) the Tutte polynomial of PSW G(n) .In this section we study and gain recursive formulas for Tn (x, y).The analytic method is based on the the relation between spanning subgraphs of G(n + 1) and spanning subgraphs of G(n).According to the second construction algorithm,we find that there exists a bijection between spanning subgraphs of G(n + 1) and spanning subgraphs of G1 , G2 , G3 inside G(n + 1),while the subgraphs G1 , G2 , G3 are isomorphic to G(n).Indeed,if H is a spanning subgraph of G(n + 1) ,we can uniquely determines three spanning subgraphs H1 , H2 and H3 of G1 , G2 and G3 ; viceversa, given three spanning subgraphs H1 , H2 and H3 of G1 , G2 and G3 , respectively, then their union provides a spanning subgraph H of the whole G(n+1). Therefore, according to the Definition 1,the Tutte polynomial of PSW G(n + 1) can be rewritten as Tn+1 (x, y) =
X
(x − 1)r(G(n+1))−r(H) (y − 1)n(H) ,
(3)
Hi ⊆Gi ,i=1,2,3
where Hi is the spanning subgraph of Gi ,and H is the union of H1 , H2 and H3 .In order to obtain the recursive formulas for Tn (x, y), we want to know the relations between r(G(n + 1)) and r(G(n)) ,r(H) and r(Hi ), n(H) and n(Hi ), for i = 1, 2, 3. It is easy to know that r(G(n+1)) = 3r(G(n))−1 and |V (H)| = |V (H1 )| + |V (H2 )| + |V (H3 )| − 3, |E(H)| = |E(H1)| + |E(H2 )| + |E(H3 )| ,for every spanning subgraph H of G(n + 1). Furthermore, two possibilities can 5
occur. 1)In the spanning subgraph Hi of Gi , the hub nodes of G(n + 1) belong to the same connected component,for any i, 1 ≤ i ≤ 3, then k(H) = k(H1 )+k(H2 )+k(H3 )−2
and
r(A) = r(H1 )+r(H2 )+r(H3 )−1
Thus n(H) = |E(H)| − r(H) = n(H1 ) + n(H2 ) + n(H3 ) + 1 r(G(n + 1)) − r(H) =
3 X
(r(G(n)) − r(Hi ))
i=1
Hence, one gets: (x − 1)r(G(n+1))−r(H) (y − 1)n(H) = (y − 1)
3 Y
(x − 1)r(G(n))−r(Hi ) (y − 1)n(Hi )
(4)
i=1
2)For certain i, 1 ≤ i ≤ 3 , in the spanning subgraph Hi , the hub nodes of G(n + 1) do not belong to the same connected component,we have k(H) = k(H1 ) + k(H2 ) + k(H3 ) − 3
and
r(A) = r(H1) + r(H2 ) + r(H3 )
Moreover n(H) = n(H1 ) + n(H2 ) + n(H3 ) Hence r(G(n + 1)) − r(H) =
3 X
(r(G(n)) − r(Hi )) − 1
i=1
Thus (x − 1)r(G(n+1))−r(H) (y − 1)n(H) =
3 Y 1 (x − 1)r(G(n))−r(Hi ) (y − 1)n(Hi ) (x − 1) i=1
(5)
Let Dn denotes the set of spanning subgraphs of G(n),we define the following partition on Dn : • D1,n denotes the set of spanning subgraphs of G(n), where the three hub nodes belong to the same connected component; C • D2,n denotes the set of spanning subgraphs of G(n), where the hub nodes A and B belong to the same connected component, and C belongs to a A B different component. Similarly, D2,n (or D2,n )denotes the set of spanning 6
subgraphs of G(n), where A(or B )does not belong to the connected component containing the other two hub nodes; • D3,n denotes the set of spanning subgraphs of G(n), where the three hub nodes belong to three different connected components. Thus, for any n ≥ 0, we have A B C Dn = D1,n ∪ D2,n ∪ D2,n ∪ D2,n ∪ D3,n
(6)
For n ≥ 0,let us define polynomial of x, y T1,n (x, y) =
X
(x − 1)r(Γn )−r(H) (y − 1)n(H)
H∈D1,n A B C Similarly,we can also define polynomials :T2,n (x, y), T2,n (x, y), T2,n (x, y), T3,n (x, y), A B C while the sum is conducted on D2,n , D2,n , D2,n , D3,n respectively.For symmetry,we have A B C T2,n (x, y) = T2,n (x, y) = T2,n (x, y),
and we can simply use T2,n (x, y) to denote one of the three polynomials. According to Definition 1 , we have: Tn (x, y) = T1,n (x, y) + 3T2,n (x, y) + T3,n (x, y)
(7)
Furthermore,we obtain the following theorem based on the relation between spanning subgraphs of G(n + 1) and spanning subgraphs of G(n). Theorem 3 For n ≥ 0, the Tutte polynomial Tn (x, y) of G(n) is given by Tn (x, y) = T1,n (x, y) + 3(x − 1)Pn (x, y) + (x − 1)2 Qn (x, y)
(8)
where T1,n (x, y), Pn (x, y),Qn (x, y) satisfy the following recurrence relation: 3 2 T1,n+1 (x, y) = (y − 1)T1,n + 3(y − 1)(x − 1)T1,n Pn
+3(x − 1)2 (y − 1)T1,n Nn2 + (x − 1)3 (y − 1)Pn3 2 2 +3(x − 1)T1,n Qn + 6T1,n Pn + 12(x − 1)T1,n Pn2 +6(x − 1)2 Pn3 + 3(x − 1)3 Pn2Qn + 6(x − 1)2 T1,n Pn Qn
Pn+1 (x, y) = 4T1,n Pn2 + 4(x − 1)T1,n Pn Qn + 4(x − 1)Pn3 +4(x − 1)2 Pn2 Qn + (x − 1)2 T1,n Q2n + (x − 1)3 Pn Q2n 7
(9)
(10)
Qn+1 (x, y) = 8Pn3 + 12(x − 1)Pn2 Qn + 6(x − 1)2 Pn Q2n + (x − 1)3 Q3n
(11)
with initial conditions T1,0 (x, y) = y + 2
P0 (x, y) = Q0 (x, y) = 1.
and T1,n ,Pn ,Qn is shorthand of T1,n (x, y), Pn (x, y),Qn (x, y) respectively. Proof: We find that x − 1 divides T2,n (x, y) and (x − 1)2 divides T3,n (x, y) for any n ≥ 0 which will be proved later.As a consequence, we can write T2,n (x, y) = (x − 1)Pn (x, y) T3,n (x, y) = (x − 1)2 Qn (x, y)
(12)
where Pn (x, y) and Qn (x, y) are polynomials of x, y. Thus we obtain Eq.(8) from Eq.(7). Now we will proof T1,n (x, y), T2,n (x, y),T2,n (x, y) satisfy the following recurrence relations which lead to the results of the theorem. 3 2 2 T1,n+1 (x, y) = (y − 1)T1,n + 3(y − 1)T1,n T2,n + 3(y − 1)T1,n T2,n 1 3 2 2 +(y − 1)T2,n + (3T1,n T3,n + 6T1,n T2,n x−1 2 3 2 +12T1,n T2,n + 6T2,n + 3T3,n T2,n + 6T1,n T2,n T3,n )
T2,n+1 (x, y) =
T3,n+1 (x, y) =
1 2 3 (4T1,n T2,n + 4T1,n T2,n T3,n + 4T2,n x−1 2 2 2 +4T2,n T3,n + T1,n T3,n + T2,n T3,n )
1 3 2 2 3 8T2,n + 12T2,n T3,n + 6T2,n T3,n + T3,n x−1
(13)
(14)
(15)
with initial conditions T1,0 (x, y) = y + 2
T2,0 (x, y) = x − 1
T3,0 (x, y) = (x − 1)2 .
and T1,n ,T2,n ,T3,n is shorthand of T1,n (x, y), T2,n (x, y),T3,n (x, y) respectively. The strategy of the proof is to study all the possible configurations of spanning subgraphs Hi in the three copies Gi of G(n) inside G(n + 1), for i = 1, 2, 3, and analyze which kind of contribution they give to T1,n+1 (x, y), T2,n+1 (x, y) and T3,n+1 (x, y). 8
G1
G2 G3
G1 +
G2 G3
G1 +
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2 G3
G1 +
G2 G3
Fig. 3. The possible configurations of spanning subgraphs Hi (i = 1, 2, 3) for T1,n+1 (x, y).The two hub nodes of Gi are connected by a solid line if they are in in the same connected component, and connected by a dotted line if they are not in the same connected component. G1
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
Fig. 4. The possible configurations of spanning subgraphs Hi (i = 1, 2, 3) for C T2,n+1 (x, y).The two hub nodes of Gi are connected by a solid line if they are in in the same connected component, and connected by a dotted line if they are not in the same connected component.
For T1,n+1 (x, y),we find it has 10 possible configurations of spanning subgraphs Hi ( i = 1, 2, 3) which is shown in Fig. 3.In the first configuration, Hi ∈ D1,n 3 for any i = 1, 2, 3. This contributes to T1,n+1 (x, y) by a term (y − 1)T1,n according to Eq.(4), since in the spanning subgraph Hi of Gi , the hub nodes of Gn+1 belong to the same connected component,for any i, 1 ≤ i ≤ 3.In the C second configuration,Hi ∈ D1,n holds for two i ∈ {1, 2, 3} and Hi ∈ D2,n holds C for the last i(for example,H1 ∈ D1,n ,H2 ∈ D1,n ,H3 ∈ D2,n ).This contributes 2 to T1,n+1 (x, y) by a term 3(y − 1)T1,n T2,n according to Eq.(4). Computing the contributions to T1,n+1 (x, y) of the 10 possible configurations and adding them together ,we obtain Eq.(13). C For T2,n+1 (x, y), we study T2,n+1 (x, y) only by symmetry.We find it has 6 possible configurations which is shown in Fig. 4.In the first configuration, H3 ∈ D1,n A B C B A ∪ D2,n ,H2 ∈ D2,n ∪ D2,n . This contributes to T2,n+1 (x, y) by a term H1 ∈ D2,n 4 2 T T according to Eq.(5), since in the spanning subgraph H1 or H2 , x−1 1,n 2,n the hub nodes of Gn+1 do not belong to the same connected component.In the A B second configuration,H3 ∈ D1,n H1 ∈ D2,n ∪ D2,n ,H2 ∈ D3,n .This contributes 4 C to T2,n+1 (x, y) by a term x−1 T1,n T2,n T3,n according to Eq.(5). Computing the C contributions to T2,n+1 (x, y) of all the 6 possible configurations and adding them together ,we obtain Eq.(14).
For T3,n+1 (x, y), we find it has 4 possible configurations which is shown in A B Fig. 5.In the first configuration, Hi ∈ D2,n ∪ D2,n ,for any i = 1, 2, 3.This 8 3 contributes to T3,n+1 (x, y) by a term x−1 T2,n according to Eq.(5), since in the spanning subgraph H1 H2 and H3 , the hub nodes of Gn+1 do not belong to A B the same connected component.In the second configuration,Hi ∈ D2,n ∪ D2,n holds for two i ∈ {1, 2, 3} and Hi ∈ D3,n holds for the last i(for example,H1 ∈ A B A B D2,n ∪ D2,n , H2 ∈ D2,n ∪ D2,n ,H3 ∈ D3,n ).This contributes to T3,n+1 (x, y) by 12 2 a term x−1 T2,n T3,n . Computing the contributions to T3,n+1 (x, y) of all the 4 possible configurations and adding them together ,we obtain Eq.(15). 9
G1
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
Fig. 5. The possible configurations of spanning subgraphs Hi (i = 1, 2, 3) for T3,n+1 (x, y).The two hub nodes of Gi are connected by a solid line if they are in in the same connected component, and connected by a dotted line if they are not in the same connected component.
For the initial conditions,It is easy to verify according to the definition. Now,we come back to proof the recurrence relations that Eqs.(9),(10),(11) show. First,we find x − 1 divides T2,n (x, y) and (x − 1)2 divides T3,n (x, y),for n ≥ 0. The results can be proved by mathematical induction based on the recurrence relations that Eqs.(14),(15) show. Inserting Eq.(12) into Eqs.(13),(14),(15) for T2,n (x, y) and T3,n (x, y),we obtain Eqs.(9),(10),(11).The initial conditions for Pn (x, y) and Qn (x, y) is easy to verified according to the initial conditions for T2,n (x, y) and T3,n (x, y). Remark:Although it is NP-hard to calculate the Tutte polynomials for general graph,the recurrence relations we obtain shows that we can calculate the Tutte polynomials for PSW with time complexity O(n) = O(log(Vn)).Thus ,we have obtain logarithmic complexity algorithm to calculate the Tutte polynomial of PSW . We can also obtain the recursive formulas for various invariants of Tutte polynomial based on their connections with the Tutte polynomial,such as the number of spanning trees,the number of connected spanning subgraphs,the number of spanning forests,the number of acyclic orientations ,the reliability polynomial and the chromatic polynomial .In this paper ,we only study the number of spanning trees and the reliability polynomial.
4
Exact result for spanning trees of PSW
Let us denote by NST (n) the number of spanning trees of PSW G(n) .According to Theorem 1 and Theorem 3,it is easy to know NST (n) = Tn (1, 1) = T1,n (1, 1) Further more , for any n ≥ 0,we have the following recurrence relation NST (n + 1) = 6NST (n)2 Pn
and
Pn+1 = 4NST (n)Pn2
where Pn is abbreviation of Pn (1, 1) ,and the initial conditions is NST (0) = T1,0 (1, 1) = 3 10
P0 = 1.
Thus NST (n) = 6NST (n − 1)2 Pn−1 (2+2)
= 6(1+2) 41 [NST (n − 2)](2×2+1) Pn−1 =... dk , 6ak 4bk [NST (n − k)]ck Pn−1 =... , 6an 4bn [NST (0)]cn P0dn
(16)
where ak , bk , ck , dk , k > 1 satisfy the following recurrence relations . ak = ak−1 + ck−1
(17)
bk = bk−1 + dk−1
(18)
ck = 2ck−1 + dk−1
(19)
dk = ck−1 + 2dk−1
(20)
with initial conditions a1 = 1
b1 = 0
c1 = 2
and
d1 = 1
Thus ck + dk = 3(ck−1 + dk−1 ) = 3k−1(c1 + d1 ) = 3k
(21)
ck − dk = ck−1 − dk−1 = c1 − d1 = 1
(22)
Then ck =
3k + 1 2
(23)
dk =
3k − 1 2
(24)
Note ak − ak−1 = ck−1
and
bk − bk−1 = dk−1
Thus k−1 X
k−1 X
k − 1 3k − 3 + ak − a1 = (ai+1 − ai ) = ci = 2 4 i=1 i=1 bk − b1 =
k−1 X
(bi+1 − bi ) =
i=1
k−1 X
di = −
i=1
11
k − 1 3k − 3 + 2 4
Hence ak =
k + 1 3k − 3 + 2 4
bk = −
(25)
k − 1 3k − 3 + 2 4
(26)
Substituting Eqs.(23),(24), (25),(26)into Eq.(16),we obtain the following result. Theorem 4 For any n ≥ 0,the number of spanning trees of PSW G(n) is given by NST (n) = 2
3n+1 −2n−3 4
3
3n+1 +2n+1 4
(27)
Note:The result coincides with the result obtained in [30],and we give an alternative approach for explicitly determining the number of spanning trees for PSW.It is smaller than the the number of spanning trees for Sierpinski gasket [32].
5
Reliability analysis of PSW
5.1 All-terminal reliability of PSW
In this section, we look upon PSW G(n) as a probabilistic graph.Each edge of G(n) has a known probability p of being operational; otherwise it is failed. Operations of different edges are statistically independent,while the nodes of G(n) never fail. The all-terminal reliability polynomial R(G(n), p) of G(n) is defined as is the probability that there is a path of operational edges between any pair of vertices of G(n). In general case ,the calculation of all-terminal reliability polynomial is NP-hard [34]. But for PSW,we obtain recursive formulas for all-terminal reliability polynomial which implies that logarithmic complexity algorithm is obtained.Further more,we get a approximate solution of R(G(n), p) based on the recursive formulas it satisfy,which shows that all terminal reliability decreases approximately as a exponential function of network order.
1 1 ),Pn 1, 1−p respeclet us simply denote by T1,n ,Pn ,the expression T1,n (1, 1−p tively for n ≥ 0, we obtain the following recurrence relations from Theorem
12
3. Tn
1 1, 1−p
T1,n+1 Pn+1
!
= T1,n
1 1, 1−p 1 1, 1−p
!
!
=
1 1, 1−p
!
(28)
p 2 T 3 + 6T1,n Pn 1 − p 1,n
(29)
= 4T1,n Pn2
(30)
with initial conditions T1,0
1 1, 1−p
!
3 − 2p = 1−p
P0
1 1, 1−p
!
= 1.
For n ≥ 0,let us simply denote by R(n) the reliability polynomial R(G(n), p) of PSW G(n) ,and define B(n) = p(Vn −2) (1 − p)(En −Vn +2) Pn
(31)
We obtain the following recurrence relation from Theorem 2 R(n + 1) = pVn+1 −1 (1 − p)En+1 −Vn+1 +1 T1,n+1 (1,
1 ) 1−p
p 3 2 T1,n + 6T1,n Pn ) 1−p 3 = p3(Vn −1) (1 − p)3(En −Vn +1) T1,n
= p3Vn −4 (1 − p)3En −3Vn +4 (
2 +6p2(Vn −1) (1 − p)2(En −Vn +1) T1,n · p(Vn −2) (1 − p)(En −Vn +2) Pn
= R(n)3 + 6R(n)2 B(n)
(32)
and B(n + 1) = p(Vn+1 −2) (1 − p)(En+1 −Vn+1 +2) Pn+1 = 4p(3Vn −5) (1 − p)(3En −3Vn +5) T1,n Pn2 = 4R(n)B(n)2
(33)
where Vn ,En denote the total number of nodes and edges of G(n) respectively and the initial conditions is R(0) = p2 (3 − 2p)
B(0) = p(1 − p)2
Note that R(n) ≫ B(n) while n → ∞,ones get 13
R(n + 1) + 2B(n + 1) = R(n)3 + 6R(n)2 B(n) + 4R(n)B(n)2 = (R(n) + 2B(n))3 − 4(R(n) + 2B(n))B(n)2 ≈ (R(n) + 2B(n))3 (34) Thus R(n) ≈ (R(n − 1) + 2B(n − 1))3 n−1
≈ (R(1) + 2B(1))3
n−1
= [p(2 − p)]3
2
≈ [p(2 − p)] 3 Vn
(35)
which shows that all terminal reliability decreases approximately as a exponential function of network order Vn .The reason we don’t use R(n−1)3 as a approximation of R(n) is that it has lager relative error than (R(n − 1) + 2B(n − 1))3 . In fact, we find that B(n) is the probability that G(n) is split into two different connected components such that one of them contains the hub nodes A,B ,the other one contains the hub node C .Thus R(n) + 2B(n) < 1,for any n ≥ 0.
5.2 Comparison of all-terminal reliability between Sierpinski gasket and PSW The Sierpinski gasket is a fractal which can be constructed iteratively [35]. The starting point is a triangle. Divide its three sides in two segments of equal length. Connect the midpoints to get four inner triangles and paint the three external ones.Apply the same process to the inner triangles but the middle one.Sierpinski gasket is the limiting set for this construction. If we look upon the Sierpinski gasket as a network,it is a deterministically growing network which has the same starting point with PSW .But the method of iteration is different from PSW [36]. The construction process of the first three generation is shown in Fig.6.We denote the Sierpinski gasket after n iterations by SG(n) with n ≥ 0.It has the same number of nodes and edges with PSW for any n ≥ 0,but the structure is quite different.For Sierpinski gasket,except the 3 outmost nodes which have degree 2, all other vertices of SG(n) have degree 4. In the large n limit, SG(n) is 4-regular.But PSW is a scale free netwokr whose degree distribution obeys power law .Thus, Sierpinski gasket and PSW are typical examples of regular network and scale free network which have the same number of nodes and edges .
✔✔❚❚
✔❚ ✔❚❚✔✔❚ ✔❚ ✔❚ ✔ ❚ ✔ ❚ ✔ ❚❚✔✔ ❚✔ ❚❚✔✔ ❚
✔❚ ✔ ❚ ✔ ❚❚✔✔ ❚
SG(0) SG(1)
SG(2)
Fig. 6. Growth process for Sierpinski gasket SG(n) from n=0 to n=2 .
14
Alfredo Donno [20] found that for each n ≥ 0, the all-terminal reliability polynomial of Sierpinski gasket SG(n) is given by R(SG(n), p) = pVn −1 (1 − p)En −Vn +1 T1,n (1,
1 ) 1−p
1 1 = T1,n 1, 1−p and with Tn 1, 1−p
T1,n+1
1 1, 1−p
!
=
p 2 T 3 + 6T1,n Nn 1 − p 1,n
(36)
!
=
p 2 T 2 Nn + T1,n Mn + 7T1,n Nn2 1 − p 1,n
(37)
!
=
3p T1,n Nn2 + 12T1,n Nn Mn + 14Nn3 1−p
(38)
Nn+1
1 1, 1−p
Mn+1
1 1, 1−p
with initial conditions T1,0
1 1, 1−p
!
3 − 2p = 1−p
N0
1 1, 1−p
!
= M0
1 1, 1−p
!
= 1.
For n ≥ 0,let us simply denote by Rs (n) the reliability polynomial R(SG(n), p) ,and define Bs (n) = p(Vn −2) (1 − p)(En −Vn +2) Nn
(39)
Ts (n) = p(Vn −3) (1 − p)(En −Vn +3) Mn
(40)
We obtain the following recurrence relation .
Rs (n + 1) = pVn+1 −1 (1 − p)En+1 −Vn+1 +1 T1,n+1 (1, = p3Vn −4 (1 − p)3En −3Vn +4 ( = Rs (n)3 + 6Rs (n)2 Bs (n)
1 ) 1−p
p 2 T 3 + 6T1,n Nn ) 1 − p 1,n (41)
and Bs (n + 1) = p(Vn+1 −2) (1 − p)(En+1 −Vn+1 +2) Nn+1 p 2 T 2 Nn + T1,n Mn + 7T1,n Nn2 ) = p(3Vn −5) (1 − p)(3En −3Vn +5) ( 1 − p 1,n = Rs (n)2 Bs (n) + Rs (n)2 Ts (n) + 7Rs (n)Bs (n)2 (42) 15
Ts (n + 1) = p(Vn+1 −3) (1 − p)(En+1 −Vn+1 +3) Mn+1 3p T1,n Nn2 + 12T1,n Nn Mn + 14Nn3 ) = p(3Vn −6) (1 − p)(3En −3Vn +6) ( 1−p = 3Rs (n)Bs (n)2 + 12Rs (n)Bs (n)Ts (n) + 14Bs (n)3 (43) with initial conditions Rs (0) = p2 (3 − 2p)
Bs (0) = p(1 − p)2
Ts (0) = (1 − p)3
Now,we compare all-terminal reliability between Sierpinski gasket and PSW while p ∈ (0, 1) . For n=0, Rs (0) = R(0)
Bs (0) = B(0)
Thus Rs (1) = R(1), But for n > 0,we can obtain from Eqs.(33) and (42) that Bs (n) > B(n). Thus ,for any n > 1,we find from Eqs.(32 )and (41) Rs (n) > R(n)
(44)
We have calculated Rs (n) and R(n) for different p ∈ (0, 1) and n by iteration.we find that Rs (n) and R(n) converge to 0 quickly with n → ∞ and Eq.(44) holds for n > 1.The results for n = 6 are shown in Fig. 7.Our results show that Sierpinski gasket is more robust than PSW against random edge failures.Thus,we obtain an example which shows that regular networks(e.g.,Sierpinski gaskets) are more robust than scale-free networks(e.g. ,PSW) against random edge failures.Whether it is true for any regular networks and scale-free networks ,is still a unresolved problem. However, recent work [37–39] have shown that inhomogeneous networks, such as scale-free networks, are more robust than homogeneous networks (e.g., exponential networks and regular networks) with respect to random deletion of nodes. Thus, combining with our above result, we can reach the following conclusion that networks (e.g.,scale-free networks) which are more robust again random node failures do not mean more robust again random breakdown of edges than those (e.g., regular lattices) which are more vulnerable to random node failures .
6
Conclusion
In this paper,we study and gain recursive formulas for the Tutte polynomial of PSW which implies that recursive formulas for various invariants of Tutte polynomial can also obtained based on their connections with the Tutte polynomial . We also obtain the rigorous solution for the the number of spanning 16
1 PSW Sierpinski gasket
All−terminal reliability of network
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.2
0.4
0.6
0.8
1
p
Fig. 7. The all-terminal reliability of Sierpinski gasket SG(6) and PSW G(6) obtained by direct calculation from Eqs.(32) and (41) .
trees of PSW by solving the recurrence relations derived from Tutte polynomial ,which give an alternative approach for explicitly determining the number of spanning trees of PSW.Further more,we analysis the all-terminal reliability of PSW based on the the recurrence relations derived from Tutte polynomial and compare the result with that of Sierpinski gasket. In contrast with the well-known conclusion that scale-free networks are more robust than homogeneous networks (e.g., exponential networks and regular networks) with respect to random deletion of nodes.Our results show that there is an example that regular networks(e.g.,Sierpinski gaskets) are more robust than scale-free networks(e.g.,PSW) against random edge failures.Whether it is true for any regular networks and scale-free networks ,is still a unresolved problem. Acknowledgment The authors are grateful to the anonymous referees for their valuable comments and suggestions. This work was supported by the National High Technology Research and Development Program(”863”Program) of China under Grant No. 2009AA01Z439.
References [1]
W.T. Tutte, 1947 Proc. Cambridge Philos. Soc. 43 26.
[2]
W.T. Tutte, 1954,Canadian. J. Math. 6 80.
17
[3]
W.T. Tutte, 1967 J. Combinatorial Theory 2 301.
[4]
J. Ellis-Monaghan and C. Merino,Graph Polynomials and Their Applications I: The Tutte Polynomial, in: “Structural Analysis of Complex Networks”(Mathhias Dehmer editor), in press; available at http://arxiv.org/abs/0803.3079v2
[5]
S.-C. Chang, L.-C. Chen and W.-S. Yang, 2007,J. Stat. Phys. 126, 3.
[6]
Z.Z.Zhang, B. Wu, Y.Lin, 2012,Physica A 391, 3342.
[7]
S.-C. Chang and L.-C. Chen, 2009,Discrete Math. Theor. Comput. Sci. 11 1.
[8]
S.-C. Chang and L.-C. Chen, 2008 Discrete Math. Theor. Comput. Sci. 10 2.
[9]
S.-C. Chang,2010, Acyclic orientations on the Sierpi´ nski gasket, available at http://arxiv.org/abs/1005.3627v1.
[10] S.-C. Chang, R. Shrock, 2003,J. Stat. Phys. 112, 815. [11] S.-C. Chang, R. Shrock,2003 ,J. Stat. Phys.112, 1019. [12] M. Rocek, R. Shrock , S,H, Tsai,1998,Physica A 252 505. [13] R. Shrock, Y. Xu,2012,J. Phys. A: Math. Theor. 45, 055212. [14] D.J.A. Welsh and C. Merino, 2000, J. Math. Phys. 41 3. [15] A. D. Sokal,2005,The multivariate Tutte polynomial (alias Potts model) for graphs and matroids,available at http://arxiv.org/abs/1005.3627v1. [16] R. Shrock,2011,J. Phys. A Math. Theor. 44, 145002. [17] J. Oxley and D.J.A. Welsh, The Tutte polynomial and percolation, in: “Graph theory and related topics”(Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), 329–339, (J.A. Bondy and U.S.R. Murty editors), Academic Press, New YorkLondon, 1979. [18] F.Jaeger, D. L.Vertigan, D. J. A.Welsh, 1990,Math. Proc.Camb. Phil. Soc. 108, 35. [19] J. Oxley,D. Welsh, 2002,Combinatorics, Probability and Computing 11, 4. [20] A. Donno, D. Iacono, The Tutte polynomial of the Sierpinski and Hanoi graphs, available at http://arxiv.org/abs/1006.5333. [21] S.-C. Chang, R. Shrock, 2001,Physica A 296, 183-233. [22] S.-C. Chang, R. Shrock, 2000,Physica A 286 ,189-238. [23] J. Salas, S.-C. Chang, R. Shrock, 2002,J. Stat. Phys. 107, 1207. [24] R. Shrock , 2000,Physica A 283,388. [25] A.-L. Barab´ asi, R. Albert 1999 Science 286, 509.
18
[26] D.J. Watts, H. Strogatz 1998 Nature (London) 393, 440. [27] S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, 2002,Phys. Rev. E 65, 066122 . [28] Z.Z.Zhang, L.L. Rong, S.G. Zhou, 2007,Physica A 377 [29] Z. Z. Zhang, S. G. Zhou, and L. C. Chen, 2007,Eur. Phys. J. B 58 337 . [30] Z.Z. Zhang, H.X. Liu, B. Wu, S.G. Zhou, 2010,Europhysics Letters 90 68002. [31] Z.Z. Zhang, Y. Qi, S.G. Zhou, W.L. Xie, J.H. Guan, 2009,Physical Review E 79 021127. [32] S.-C. Chang, L.-C. Chen and W.-S. Yang, 2007,J. Stat. Phys. 126, 649. [33] E. M. Bollt, D. ben-Avraham, 2005,New J. Phys. 7, 26 . [34] M.O. Ball, 1980,Networks 10 153 . [35] B.Mandelbrot, 1982, The Fractal Geometry of Nature Freeman ,San Francisco. [36] R.Hilfer and A.Blumen, 1984,J. Phys. A: Math. Gen. 17 L537-L545. [37] R. Albert, H. Jeong, A.-L. Barab´ asi,2000,Nature (London) 406, 378 . [38] D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, 2000,Phys. Rev. Lett. 85, 5468 . [39] R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, 2001,Phys. Rev. Lett. 86, 3682 .
19
Tutte polynomial of pseudofractal scale-free web Junhao Peng a,b,c Guoai Xu b a School
of Mathematics and Information Science, Guangzhou University , Guangzhou 510006 , China.
b State
Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications , Beijing 100876 ,China.
c Key
Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University,Guangzhou 510006 ,China.
Abstract The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both combinatorics and statistical physics. It contains various numerical invariants and polynomial invariants ,such as the number of spanning trees,the number of spanning forests , the number of acyclic orientations , the reliability polynomial,chromatic polynomial and flow polynomial . In this paper,we study and gain recursive formulas for the Tutte polynomial of pseudofractal scale-free web(PSW) which implies logarithmic complexity algorithm is obtained to calculate the Tutte polynomial of PSW although it is NP-hard for general graph.We also obtain the rigorous solution for the the number of spanning trees of PSW by solving the recurrence relations derived from Tutte polynomial ,which give an alternative approach for explicitly determining the number of spanning trees of PSW.Further more,we analysis the all-terminal reliability of PSW and compare the results with that of Sierpinski gasket which has the same number of nodes and edges with PSW. In contrast with the well-known conclusion that scale-free networks are more robust against removal of nodes than homogeneous networks (e.g., exponential networks and regular networks).Our results show that Sierpinski gasket (which is a regular network) are more robust against random edge failures than PSW (which is a scalefree network) .Whether it is true for any regular networks and scale-free networks ,is still a unresolved problem. Key words: Pseudofractal scale-free web, Tutte polynomial, Spanning trees, Reliability polynomial PACS: 05.45.Df, 89.75.Hc, 05.10.-a
Email address: [email protected] (Junhao Peng).
Preprint submitted to Physica A
2 May 2014
1
Introduction
The Tutte polynomial is a two-variable polynomial which can be associated with a graph, a matrix, or, more generally, with a matroid . This polynomial was introduced by W.T. Tutte [1–3] and has many interesting applications in several areas of sciences such as combinatorics, probability, statistical mechanics and computer science. It is quite interesting since several combinatorial, enumerative and algebraic properties of the graph can be investigated by considering special evaluations of it [4]. For instance, one gets information about the number of spanning trees [5, 6], spanning connected subgraphs [7], spanning forests [8] and acyclic orientations [9] of the graph by evaluating tutte polynomial at particular points (x, y) . Moreover, the Tutte polynomial contains several other polynomial invariants, such as flow polynomial [10] , reliability polynomial [11] and chromatic polynomial [2, 12, 13]. It has also many interesting connections with statistical mechanical models such as the Potts model [14–16] and the percolation [17]. Despite its ubiquity,there are no widely-available effective computational tools to compute the Tutte polynomials of a general graph of reasonable size.It is shown that, many of the relevant coefficients do not even have good randomised approximation schemes and various decision problems based on the coefficients are NP-hard [18,19].Although it is hard to to compute the Tutte polynomials,a lot of efforts have been devoted to the study the Tutte polynomials of different graph such as polygon chain graphs [16], Sierpinski gaskets [20] and strips of lattices [21–24].However, polygon chain ,lattices and Sierpinski gaskets cannot well mimic the real-life networks, which have been recently found to synchronously exhibit two striking properties: scale-free behavior [25] and small-world effects [26]. Pseudofractal scale-free web(PSW) we studied is a deterministically growing network introduced by S.N. Dorogovtsev [27] which is used to model scale-free network with small-world effect.Lots of job was devoted to study its properties ,such as degree distribution ,degree correlation , clustering coefficient [27, 28] ,diameter [28],average path length [29], the number of spanning trees [30] and mean first-passage time for random walk [31].As for its tutte polynomial and reliability polynomial ,to the best of our knowledge, related research was rarely reported . In this paper,we study and gain recursive formulas for the Tutte polynomial of PSW .The analytic method is based on the its recursive construction and self-similar structure .Recursive formulas for various invariants of Tutte polynomial can also obtained based on their connections with the Tutte polynomial . We also obtain the rigorous solution for the the number of spanning trees by solving the recurrence relations derived from the Tutte polynomial,which coincides with the result obtained in [30] .Thus we give an alternative approach for explicitly determining the number of spanning trees of PSW .Further more,we 2
analysis the all-terminal reliability of PSW and compare the results with that of Sierpinski gasket which has the same number of nodes and edges with PSW. In contrast with the well-known conclusion that scale-free networks are more robust against random node failures than homogeneous networks (e.g., exponential networks and regular networks) .Our results show that Sierpinski gasket (which is a regular network) are more robust against random edge failures than PSW (which is a scale-free network) .
2
Preliminaries
2.1 The Tutte polynomial Let G = (V (G), E(G)) denotes a graph with vertex set V (G) and edge set E(G); we will often write V and E, when there is no risk of confusion, and so G = (V, E). A subgraph H = (V (H), E(H)) of a graph G = (V (G), E(G)) is said spanning if the condition V (H) = V (G) is satisfied. In particular, a spanning tree of G is a spanning subgraph of G which is a tree.Let H be a spanning subgraph of G and k(H) be the number of connected components of H.then the rank r(H) and the nullity n(H) of H are defined as r(H) = |V (H)| − k(H) = |V (H)| − k(H) n(H) = |E(H)| − r(H) = |E(H)| − |V (H)| + k(H)
Definition 1 Let G = (V, E) be a graph. The Tutte polynomial T (G; x, y) of G is defined as [4] T (G; x, y) =
X
(x − 1)r(G)−r(H) (y − 1)n(H)
(1)
H⊆G
where the sum runs over all the spanning subgraphs H of G . The Tutte polynomial can be evaluated at particular points (x, y) to give numerical invariants,such as the number of spanning trees, the number of forests and the number of connected spanning subgraphs.The following theorem [4] depicts the relations between the Tutte polynomial and its numerical invariants. Theorem 1 Let G = (V, E) be a connected graph ,Then: (1) T (G; 1, 1) is the number of spanning trees; (2) T (G; 1, 2) is the number of spanning connected subgraphs of G; 3
Fig. 1. Growth process for PSW from n=0 to n=2
(3) T (G; 2, 1) is the number of spanning forests of G; The Tutte polynomial also has a variety of single-variable polynomial invariants associated with the graph,such as the all-terminal reliability polynomial,flow polynomial and the chromatic polynomial.In this paper we only study the all-terminal reliability polynomial. Let G = (V, E) be a graph,each edge of G has a known probability p of being operational; otherwise it is failed. Operations of different edges are statistically independent,while the nodes of G never fail. The all-terminal reliability polynomial R(G, p) of G is defined as is the probability that there is a path of operational edges between any pair of vertices of G.The connection between the Tutte polynomial and all-terminal reliability polynomial is given by the following theorem [4]. Theorem 2 Let G = (V, E) be a graph, Then R(G, p) = p
|V (G)|−1
(1 − p)
|E(G)|−|V (G)|+1
1 T G; 1, 1−p
!
(2)
2.2 Structure of PSW The scale-free network we studied is a deterministically growing network which can be constructed iteratively [27] . We denote the pseudofractal scale-free web(PSW) after n iterations by G(n) with n ≥ 0. Then it is constructed as follows: For n = 0, G0 is a triangle. For n ≥ 1, Gn is obtained from Gn−1 : every existing edge in Gn−1 introduces a new node connected to both ends of the edge. The construction process of the first three generation is shown in Fig. 1. The PSW exhibits some typical properties of real networks. Its degree distribution P (k) obeys a power law P (k) ∼ k 1+ln 3/ ln 2 [27], the average path length scales logarithmically with network order [29]. The network also has an equivalent construction method [30, 33],as can be seen in Fig. 2 : to obtain G(n + 1), one can make three copies of G(n) and join them at the three most 4
C
B A G1
C
C G2
A B
G1
B A
G2
A
G3
B G3
C
Fig. 2. Second construction method of PSW: G(n + 1) is composed of three copies of G(n) denoted as G1 ,G2 ,G3 ,the three hub nodes of which are represented by A,B,C in the corresponding triangle(The left side of the figure ).In the merging process,hub node A of G1 and hub node B of G3 ,hub node A of G3 and hub node B of G2 , hub node A of G2 and hub node B of G1 are identified as a hub node A,B,C of G(n + 1) respectively(The right side of the figure ).
connected nodes denoted by A,B,C, which are called the hub nodes of G(n+1) in this paper. According to the second construction algorithm, one can see that at each step n, the total number of edges in the network increases by a factor of 3. Thus, the total number of edges for G(n) is En = 3n+1 .We can also find n+1 that the total number of nodes for G(n) is Vn = 3 2 +3 .
3
Tutte polynomial of PSW
Let us simply denote by Tn (x, y) the Tutte polynomial of PSW G(n) .In this section we study and gain recursive formulas for Tn (x, y).The analytic method is based on the the relation between spanning subgraphs of G(n + 1) and spanning subgraphs of G(n).According to the second construction algorithm,we find that there exists a bijection between spanning subgraphs of G(n + 1) and spanning subgraphs of G1 , G2 , G3 inside G(n + 1),while the subgraphs G1 , G2 , G3 are isomorphic to G(n).Indeed,if H is a spanning subgraph of G(n + 1) ,we can uniquely determines three spanning subgraphs H1 , H2 and H3 of G1 , G2 and G3 ; viceversa, given three spanning subgraphs H1 , H2 and H3 of G1 , G2 and G3 , respectively, then their union provides a spanning subgraph H of the whole G(n+1). Therefore, according to the Definition 1,the Tutte polynomial of PSW G(n + 1) can be rewritten as Tn+1 (x, y) =
X
(x − 1)r(G(n+1))−r(H) (y − 1)n(H) ,
(3)
Hi ⊆Gi ,i=1,2,3
where Hi is the spanning subgraph of Gi ,and H is the union of H1 , H2 and H3 .In order to obtain the recursive formulas for Tn (x, y), we want to know the relations between r(G(n + 1)) and r(G(n)) ,r(H) and r(Hi ), n(H) and n(Hi ), for i = 1, 2, 3. It is easy to know that r(G(n+1)) = 3r(G(n))−1 and |V (H)| = |V (H1 )| + |V (H2 )| + |V (H3 )| − 3, |E(H)| = |E(H1)| + |E(H2 )| + |E(H3 )| ,for every spanning subgraph H of G(n + 1). Furthermore, two possibilities can 5
occur. 1)In the spanning subgraph Hi of Gi , the hub nodes of G(n + 1) belong to the same connected component,for any i, 1 ≤ i ≤ 3, then k(H) = k(H1 )+k(H2 )+k(H3 )−2
and
r(A) = r(H1 )+r(H2 )+r(H3 )−1
Thus n(H) = |E(H)| − r(H) = n(H1 ) + n(H2 ) + n(H3 ) + 1 r(G(n + 1)) − r(H) =
3 X
(r(G(n)) − r(Hi ))
i=1
Hence, one gets: (x − 1)r(G(n+1))−r(H) (y − 1)n(H) = (y − 1)
3 Y
(x − 1)r(G(n))−r(Hi ) (y − 1)n(Hi )
(4)
i=1
2)For certain i, 1 ≤ i ≤ 3 , in the spanning subgraph Hi , the hub nodes of G(n + 1) do not belong to the same connected component,we have k(H) = k(H1 ) + k(H2 ) + k(H3 ) − 3
and
r(A) = r(H1) + r(H2 ) + r(H3 )
Moreover n(H) = n(H1 ) + n(H2 ) + n(H3 ) Hence r(G(n + 1)) − r(H) =
3 X
(r(G(n)) − r(Hi )) − 1
i=1
Thus (x − 1)r(G(n+1))−r(H) (y − 1)n(H) =
3 Y 1 (x − 1)r(G(n))−r(Hi ) (y − 1)n(Hi ) (x − 1) i=1
(5)
Let Dn denotes the set of spanning subgraphs of G(n),we define the following partition on Dn : • D1,n denotes the set of spanning subgraphs of G(n), where the three hub nodes belong to the same connected component; C • D2,n denotes the set of spanning subgraphs of G(n), where the hub nodes A and B belong to the same connected component, and C belongs to a A B different component. Similarly, D2,n (or D2,n )denotes the set of spanning 6
subgraphs of G(n), where A(or B )does not belong to the connected component containing the other two hub nodes; • D3,n denotes the set of spanning subgraphs of G(n), where the three hub nodes belong to three different connected components. Thus, for any n ≥ 0, we have A B C Dn = D1,n ∪ D2,n ∪ D2,n ∪ D2,n ∪ D3,n
(6)
For n ≥ 0,let us define polynomial of x, y T1,n (x, y) =
X
(x − 1)r(Γn )−r(H) (y − 1)n(H)
H∈D1,n A B C Similarly,we can also define polynomials :T2,n (x, y), T2,n (x, y), T2,n (x, y), T3,n (x, y), A B C while the sum is conducted on D2,n , D2,n , D2,n , D3,n respectively.For symmetry,we have A B C T2,n (x, y) = T2,n (x, y) = T2,n (x, y),
and we can simply use T2,n (x, y) to denote one of the three polynomials. According to Definition 1 , we have: Tn (x, y) = T1,n (x, y) + 3T2,n (x, y) + T3,n (x, y)
(7)
Furthermore,we obtain the following theorem based on the relation between spanning subgraphs of G(n + 1) and spanning subgraphs of G(n). Theorem 3 For n ≥ 0, the Tutte polynomial Tn (x, y) of G(n) is given by Tn (x, y) = T1,n (x, y) + 3(x − 1)Pn (x, y) + (x − 1)2 Qn (x, y)
(8)
where T1,n (x, y), Pn (x, y),Qn (x, y) satisfy the following recurrence relation: 3 2 T1,n+1 (x, y) = (y − 1)T1,n + 3(y − 1)(x − 1)T1,n Pn
+3(x − 1)2 (y − 1)T1,n Nn2 + (x − 1)3 (y − 1)Pn3 2 2 +3(x − 1)T1,n Qn + 6T1,n Pn + 12(x − 1)T1,n Pn2 +6(x − 1)2 Pn3 + 3(x − 1)3 Pn2Qn + 6(x − 1)2 T1,n Pn Qn
Pn+1 (x, y) = 4T1,n Pn2 + 4(x − 1)T1,n Pn Qn + 4(x − 1)Pn3 +4(x − 1)2 Pn2 Qn + (x − 1)2 T1,n Q2n + (x − 1)3 Pn Q2n 7
(9)
(10)
Qn+1 (x, y) = 8Pn3 + 12(x − 1)Pn2 Qn + 6(x − 1)2 Pn Q2n + (x − 1)3 Q3n
(11)
with initial conditions T1,0 (x, y) = y + 2
P0 (x, y) = Q0 (x, y) = 1.
and T1,n ,Pn ,Qn is shorthand of T1,n (x, y), Pn (x, y),Qn (x, y) respectively. Proof: We find that x − 1 divides T2,n (x, y) and (x − 1)2 divides T3,n (x, y) for any n ≥ 0 which will be proved later.As a consequence, we can write T2,n (x, y) = (x − 1)Pn (x, y) T3,n (x, y) = (x − 1)2 Qn (x, y)
(12)
where Pn (x, y) and Qn (x, y) are polynomials of x, y. Thus we obtain Eq.(8) from Eq.(7). Now we will proof T1,n (x, y), T2,n (x, y),T2,n (x, y) satisfy the following recurrence relations which lead to the results of the theorem. 3 2 2 T1,n+1 (x, y) = (y − 1)T1,n + 3(y − 1)T1,n T2,n + 3(y − 1)T1,n T2,n 1 3 2 2 +(y − 1)T2,n + (3T1,n T3,n + 6T1,n T2,n x−1 2 3 2 +12T1,n T2,n + 6T2,n + 3T3,n T2,n + 6T1,n T2,n T3,n )
T2,n+1 (x, y) =
T3,n+1 (x, y) =
1 2 3 (4T1,n T2,n + 4T1,n T2,n T3,n + 4T2,n x−1 2 2 2 +4T2,n T3,n + T1,n T3,n + T2,n T3,n )
1 3 2 2 3 8T2,n + 12T2,n T3,n + 6T2,n T3,n + T3,n x−1
(13)
(14)
(15)
with initial conditions T1,0 (x, y) = y + 2
T2,0 (x, y) = x − 1
T3,0 (x, y) = (x − 1)2 .
and T1,n ,T2,n ,T3,n is shorthand of T1,n (x, y), T2,n (x, y),T3,n (x, y) respectively. The strategy of the proof is to study all the possible configurations of spanning subgraphs Hi in the three copies Gi of G(n) inside G(n + 1), for i = 1, 2, 3, and analyze which kind of contribution they give to T1,n+1 (x, y), T2,n+1 (x, y) and T3,n+1 (x, y). 8
G1
G2 G3
G1 +
G2 G3
G1 +
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2
G1 +
G3
G2 G3
G1 +
G2 G3
Fig. 3. The possible configurations of spanning subgraphs Hi (i = 1, 2, 3) for T1,n+1 (x, y).The two hub nodes of Gi are connected by a solid line if they are in in the same connected component, and connected by a dotted line if they are not in the same connected component. G1
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
Fig. 4. The possible configurations of spanning subgraphs Hi (i = 1, 2, 3) for C T2,n+1 (x, y).The two hub nodes of Gi are connected by a solid line if they are in in the same connected component, and connected by a dotted line if they are not in the same connected component.
For T1,n+1 (x, y),we find it has 10 possible configurations of spanning subgraphs Hi ( i = 1, 2, 3) which is shown in Fig. 3.In the first configuration, Hi ∈ D1,n 3 for any i = 1, 2, 3. This contributes to T1,n+1 (x, y) by a term (y − 1)T1,n according to Eq.(4), since in the spanning subgraph Hi of Gi , the hub nodes of Gn+1 belong to the same connected component,for any i, 1 ≤ i ≤ 3.In the C second configuration,Hi ∈ D1,n holds for two i ∈ {1, 2, 3} and Hi ∈ D2,n holds C for the last i(for example,H1 ∈ D1,n ,H2 ∈ D1,n ,H3 ∈ D2,n ).This contributes 2 to T1,n+1 (x, y) by a term 3(y − 1)T1,n T2,n according to Eq.(4). Computing the contributions to T1,n+1 (x, y) of the 10 possible configurations and adding them together ,we obtain Eq.(13). C For T2,n+1 (x, y), we study T2,n+1 (x, y) only by symmetry.We find it has 6 possible configurations which is shown in Fig. 4.In the first configuration, H3 ∈ D1,n A B C B A ∪ D2,n ,H2 ∈ D2,n ∪ D2,n . This contributes to T2,n+1 (x, y) by a term H1 ∈ D2,n 4 2 T T according to Eq.(5), since in the spanning subgraph H1 or H2 , x−1 1,n 2,n the hub nodes of Gn+1 do not belong to the same connected component.In the A B second configuration,H3 ∈ D1,n H1 ∈ D2,n ∪ D2,n ,H2 ∈ D3,n .This contributes 4 C to T2,n+1 (x, y) by a term x−1 T1,n T2,n T3,n according to Eq.(5). Computing the C contributions to T2,n+1 (x, y) of all the 6 possible configurations and adding them together ,we obtain Eq.(14).
For T3,n+1 (x, y), we find it has 4 possible configurations which is shown in A B Fig. 5.In the first configuration, Hi ∈ D2,n ∪ D2,n ,for any i = 1, 2, 3.This 8 3 contributes to T3,n+1 (x, y) by a term x−1 T2,n according to Eq.(5), since in the spanning subgraph H1 H2 and H3 , the hub nodes of Gn+1 do not belong to A B the same connected component.In the second configuration,Hi ∈ D2,n ∪ D2,n holds for two i ∈ {1, 2, 3} and Hi ∈ D3,n holds for the last i(for example,H1 ∈ A B A B D2,n ∪ D2,n , H2 ∈ D2,n ∪ D2,n ,H3 ∈ D3,n ).This contributes to T3,n+1 (x, y) by 12 2 a term x−1 T2,n T3,n . Computing the contributions to T3,n+1 (x, y) of all the 4 possible configurations and adding them together ,we obtain Eq.(15). 9
G1
G2 G3
G1 +
G2 G3
G1 +
G2 G3
G1 +
G2 G3
Fig. 5. The possible configurations of spanning subgraphs Hi (i = 1, 2, 3) for T3,n+1 (x, y).The two hub nodes of Gi are connected by a solid line if they are in in the same connected component, and connected by a dotted line if they are not in the same connected component.
For the initial conditions,It is easy to verify according to the definition. Now,we come back to proof the recurrence relations that Eqs.(9),(10),(11) show. First,we find x − 1 divides T2,n (x, y) and (x − 1)2 divides T3,n (x, y),for n ≥ 0. The results can be proved by mathematical induction based on the recurrence relations that Eqs.(14),(15) show. Inserting Eq.(12) into Eqs.(13),(14),(15) for T2,n (x, y) and T3,n (x, y),we obtain Eqs.(9),(10),(11).The initial conditions for Pn (x, y) and Qn (x, y) is easy to verified according to the initial conditions for T2,n (x, y) and T3,n (x, y). Remark:Although it is NP-hard to calculate the Tutte polynomials for general graph,the recurrence relations we obtain shows that we can calculate the Tutte polynomials for PSW with time complexity O(n) = O(log(Vn)).Thus ,we have obtain logarithmic complexity algorithm to calculate the Tutte polynomial of PSW . We can also obtain the recursive formulas for various invariants of Tutte polynomial based on their connections with the Tutte polynomial,such as the number of spanning trees,the number of connected spanning subgraphs,the number of spanning forests,the number of acyclic orientations ,the reliability polynomial and the chromatic polynomial .In this paper ,we only study the number of spanning trees and the reliability polynomial.
4
Exact result for spanning trees of PSW
Let us denote by NST (n) the number of spanning trees of PSW G(n) .According to Theorem 1 and Theorem 3,it is easy to know NST (n) = Tn (1, 1) = T1,n (1, 1) Further more , for any n ≥ 0,we have the following recurrence relation NST (n + 1) = 6NST (n)2 Pn
and
Pn+1 = 4NST (n)Pn2
where Pn is abbreviation of Pn (1, 1) ,and the initial conditions is NST (0) = T1,0 (1, 1) = 3 10
P0 = 1.
Thus NST (n) = 6NST (n − 1)2 Pn−1 (2+2)
= 6(1+2) 41 [NST (n − 2)](2×2+1) Pn−1 =... dk , 6ak 4bk [NST (n − k)]ck Pn−1 =... , 6an 4bn [NST (0)]cn P0dn
(16)
where ak , bk , ck , dk , k > 1 satisfy the following recurrence relations . ak = ak−1 + ck−1
(17)
bk = bk−1 + dk−1
(18)
ck = 2ck−1 + dk−1
(19)
dk = ck−1 + 2dk−1
(20)
with initial conditions a1 = 1
b1 = 0
c1 = 2
and
d1 = 1
Thus ck + dk = 3(ck−1 + dk−1 ) = 3k−1(c1 + d1 ) = 3k
(21)
ck − dk = ck−1 − dk−1 = c1 − d1 = 1
(22)
Then ck =
3k + 1 2
(23)
dk =
3k − 1 2
(24)
Note ak − ak−1 = ck−1
and
bk − bk−1 = dk−1
Thus k−1 X
k−1 X
k − 1 3k − 3 + ak − a1 = (ai+1 − ai ) = ci = 2 4 i=1 i=1 bk − b1 =
k−1 X
(bi+1 − bi ) =
i=1
k−1 X
di = −
i=1
11
k − 1 3k − 3 + 2 4
Hence ak =
k + 1 3k − 3 + 2 4
bk = −
(25)
k − 1 3k − 3 + 2 4
(26)
Substituting Eqs.(23),(24), (25),(26)into Eq.(16),we obtain the following result. Theorem 4 For any n ≥ 0,the number of spanning trees of PSW G(n) is given by NST (n) = 2
3n+1 −2n−3 4
3
3n+1 +2n+1 4
(27)
Note:The result coincides with the result obtained in [30],and we give an alternative approach for explicitly determining the number of spanning trees for PSW.It is smaller than the the number of spanning trees for Sierpinski gasket [32].
5
Reliability analysis of PSW
5.1 All-terminal reliability of PSW
In this section, we look upon PSW G(n) as a probabilistic graph.Each edge of G(n) has a known probability p of being operational; otherwise it is failed. Operations of different edges are statistically independent,while the nodes of G(n) never fail. The all-terminal reliability polynomial R(G(n), p) of G(n) is defined as is the probability that there is a path of operational edges between any pair of vertices of G(n). In general case ,the calculation of all-terminal reliability polynomial is NP-hard [34]. But for PSW,we obtain recursive formulas for all-terminal reliability polynomial which implies that logarithmic complexity algorithm is obtained.Further more,we get a approximate solution of R(G(n), p) based on the recursive formulas it satisfy,which shows that all terminal reliability decreases approximately as a exponential function of network order.
1 1 ),Pn 1, 1−p respeclet us simply denote by T1,n ,Pn ,the expression T1,n (1, 1−p tively for n ≥ 0, we obtain the following recurrence relations from Theorem
12
3. Tn
1 1, 1−p
T1,n+1 Pn+1
!
= T1,n
1 1, 1−p 1 1, 1−p
!
!
=
1 1, 1−p
!
(28)
p 2 T 3 + 6T1,n Pn 1 − p 1,n
(29)
= 4T1,n Pn2
(30)
with initial conditions T1,0
1 1, 1−p
!
3 − 2p = 1−p
P0
1 1, 1−p
!
= 1.
For n ≥ 0,let us simply denote by R(n) the reliability polynomial R(G(n), p) of PSW G(n) ,and define B(n) = p(Vn −2) (1 − p)(En −Vn +2) Pn
(31)
We obtain the following recurrence relation from Theorem 2 R(n + 1) = pVn+1 −1 (1 − p)En+1 −Vn+1 +1 T1,n+1 (1,
1 ) 1−p
p 3 2 T1,n + 6T1,n Pn ) 1−p 3 = p3(Vn −1) (1 − p)3(En −Vn +1) T1,n
= p3Vn −4 (1 − p)3En −3Vn +4 (
2 +6p2(Vn −1) (1 − p)2(En −Vn +1) T1,n · p(Vn −2) (1 − p)(En −Vn +2) Pn
= R(n)3 + 6R(n)2 B(n)
(32)
and B(n + 1) = p(Vn+1 −2) (1 − p)(En+1 −Vn+1 +2) Pn+1 = 4p(3Vn −5) (1 − p)(3En −3Vn +5) T1,n Pn2 = 4R(n)B(n)2
(33)
where Vn ,En denote the total number of nodes and edges of G(n) respectively and the initial conditions is R(0) = p2 (3 − 2p)
B(0) = p(1 − p)2
Note that R(n) ≫ B(n) while n → ∞,ones get 13
R(n + 1) + 2B(n + 1) = R(n)3 + 6R(n)2 B(n) + 4R(n)B(n)2 = (R(n) + 2B(n))3 − 4(R(n) + 2B(n))B(n)2 ≈ (R(n) + 2B(n))3 (34) Thus R(n) ≈ (R(n − 1) + 2B(n − 1))3 n−1
≈ (R(1) + 2B(1))3
n−1
= [p(2 − p)]3
2
≈ [p(2 − p)] 3 Vn
(35)
which shows that all terminal reliability decreases approximately as a exponential function of network order Vn .The reason we don’t use R(n−1)3 as a approximation of R(n) is that it has lager relative error than (R(n − 1) + 2B(n − 1))3 . In fact, we find that B(n) is the probability that G(n) is split into two different connected components such that one of them contains the hub nodes A,B ,the other one contains the hub node C .Thus R(n) + 2B(n) < 1,for any n ≥ 0.
5.2 Comparison of all-terminal reliability between Sierpinski gasket and PSW The Sierpinski gasket is a fractal which can be constructed iteratively [35]. The starting point is a triangle. Divide its three sides in two segments of equal length. Connect the midpoints to get four inner triangles and paint the three external ones.Apply the same process to the inner triangles but the middle one.Sierpinski gasket is the limiting set for this construction. If we look upon the Sierpinski gasket as a network,it is a deterministically growing network which has the same starting point with PSW .But the method of iteration is different from PSW [36]. The construction process of the first three generation is shown in Fig.6.We denote the Sierpinski gasket after n iterations by SG(n) with n ≥ 0.It has the same number of nodes and edges with PSW for any n ≥ 0,but the structure is quite different.For Sierpinski gasket,except the 3 outmost nodes which have degree 2, all other vertices of SG(n) have degree 4. In the large n limit, SG(n) is 4-regular.But PSW is a scale free netwokr whose degree distribution obeys power law .Thus, Sierpinski gasket and PSW are typical examples of regular network and scale free network which have the same number of nodes and edges .
✔✔❚❚
✔❚ ✔❚❚✔✔❚ ✔❚ ✔❚ ✔ ❚ ✔ ❚ ✔ ❚❚✔✔ ❚✔ ❚❚✔✔ ❚
✔❚ ✔ ❚ ✔ ❚❚✔✔ ❚
SG(0) SG(1)
SG(2)
Fig. 6. Growth process for Sierpinski gasket SG(n) from n=0 to n=2 .
14
Alfredo Donno [20] found that for each n ≥ 0, the all-terminal reliability polynomial of Sierpinski gasket SG(n) is given by R(SG(n), p) = pVn −1 (1 − p)En −Vn +1 T1,n (1,
1 ) 1−p
1 1 = T1,n 1, 1−p and with Tn 1, 1−p
T1,n+1
1 1, 1−p
!
=
p 2 T 3 + 6T1,n Nn 1 − p 1,n
(36)
!
=
p 2 T 2 Nn + T1,n Mn + 7T1,n Nn2 1 − p 1,n
(37)
!
=
3p T1,n Nn2 + 12T1,n Nn Mn + 14Nn3 1−p
(38)
Nn+1
1 1, 1−p
Mn+1
1 1, 1−p
with initial conditions T1,0
1 1, 1−p
!
3 − 2p = 1−p
N0
1 1, 1−p
!
= M0
1 1, 1−p
!
= 1.
For n ≥ 0,let us simply denote by Rs (n) the reliability polynomial R(SG(n), p) ,and define Bs (n) = p(Vn −2) (1 − p)(En −Vn +2) Nn
(39)
Ts (n) = p(Vn −3) (1 − p)(En −Vn +3) Mn
(40)
We obtain the following recurrence relation .
Rs (n + 1) = pVn+1 −1 (1 − p)En+1 −Vn+1 +1 T1,n+1 (1, = p3Vn −4 (1 − p)3En −3Vn +4 ( = Rs (n)3 + 6Rs (n)2 Bs (n)
1 ) 1−p
p 2 T 3 + 6T1,n Nn ) 1 − p 1,n (41)
and Bs (n + 1) = p(Vn+1 −2) (1 − p)(En+1 −Vn+1 +2) Nn+1 p 2 T 2 Nn + T1,n Mn + 7T1,n Nn2 ) = p(3Vn −5) (1 − p)(3En −3Vn +5) ( 1 − p 1,n = Rs (n)2 Bs (n) + Rs (n)2 Ts (n) + 7Rs (n)Bs (n)2 (42) 15
Ts (n + 1) = p(Vn+1 −3) (1 − p)(En+1 −Vn+1 +3) Mn+1 3p T1,n Nn2 + 12T1,n Nn Mn + 14Nn3 ) = p(3Vn −6) (1 − p)(3En −3Vn +6) ( 1−p = 3Rs (n)Bs (n)2 + 12Rs (n)Bs (n)Ts (n) + 14Bs (n)3 (43) with initial conditions Rs (0) = p2 (3 − 2p)
Bs (0) = p(1 − p)2
Ts (0) = (1 − p)3
Now,we compare all-terminal reliability between Sierpinski gasket and PSW while p ∈ (0, 1) . For n=0, Rs (0) = R(0)
Bs (0) = B(0)
Thus Rs (1) = R(1), But for n > 0,we can obtain from Eqs.(33) and (42) that Bs (n) > B(n). Thus ,for any n > 1,we find from Eqs.(32 )and (41) Rs (n) > R(n)
(44)
We have calculated Rs (n) and R(n) for different p ∈ (0, 1) and n by iteration.we find that Rs (n) and R(n) converge to 0 quickly with n → ∞ and Eq.(44) holds for n > 1.The results for n = 6 are shown in Fig. 7.Our results show that Sierpinski gasket is more robust than PSW against random edge failures.Thus,we obtain an example which shows that regular networks(e.g.,Sierpinski gaskets) are more robust than scale-free networks(e.g. ,PSW) against random edge failures.Whether it is true for any regular networks and scale-free networks ,is still a unresolved problem. However, recent work [37–39] have shown that inhomogeneous networks, such as scale-free networks, are more robust than homogeneous networks (e.g., exponential networks and regular networks) with respect to random deletion of nodes. Thus, combining with our above result, we can reach the following conclusion that networks (e.g.,scale-free networks) which are more robust again random node failures do not mean more robust again random breakdown of edges than those (e.g., regular lattices) which are more vulnerable to random node failures .
6
Conclusion
In this paper,we study and gain recursive formulas for the Tutte polynomial of PSW which implies that recursive formulas for various invariants of Tutte polynomial can also obtained based on their connections with the Tutte polynomial . We also obtain the rigorous solution for the the number of spanning 16
1 PSW Sierpinski gasket
All−terminal reliability of network
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.2
0.4
0.6
0.8
1
p
Fig. 7. The all-terminal reliability of Sierpinski gasket SG(6) and PSW G(6) obtained by direct calculation from Eqs.(32) and (41) .
trees of PSW by solving the recurrence relations derived from Tutte polynomial ,which give an alternative approach for explicitly determining the number of spanning trees of PSW.Further more,we analysis the all-terminal reliability of PSW based on the the recurrence relations derived from Tutte polynomial and compare the result with that of Sierpinski gasket. In contrast with the well-known conclusion that scale-free networks are more robust than homogeneous networks (e.g., exponential networks and regular networks) with respect to random deletion of nodes.Our results show that there is an example that regular networks(e.g.,Sierpinski gaskets) are more robust than scale-free networks(e.g.,PSW) against random edge failures.Whether it is true for any regular networks and scale-free networks ,is still a unresolved problem. Acknowledgment The authors are grateful to the anonymous referees for their valuable comments and suggestions. This work was supported by the National High Technology Research and Development Program(”863”Program) of China under Grant No. 2009AA01Z439.
References [1]
W.T. Tutte, 1947 Proc. Cambridge Philos. Soc. 43 26.
[2]
W.T. Tutte, 1954,Canadian. J. Math. 6 80.
17
[3]
W.T. Tutte, 1967 J. Combinatorial Theory 2 301.
[4]
J. Ellis-Monaghan and C. Merino,Graph Polynomials and Their Applications I: The Tutte Polynomial, in: “Structural Analysis of Complex Networks”(Mathhias Dehmer editor), in press; available at http://arxiv.org/abs/0803.3079v2
[5]
S.-C. Chang, L.-C. Chen and W.-S. Yang, 2007,J. Stat. Phys. 126, 3.
[6]
Z.Z.Zhang, B. Wu, Y.Lin, 2012,Physica A 391, 3342.
[7]
S.-C. Chang and L.-C. Chen, 2009,Discrete Math. Theor. Comput. Sci. 11 1.
[8]
S.-C. Chang and L.-C. Chen, 2008 Discrete Math. Theor. Comput. Sci. 10 2.
[9]
S.-C. Chang,2010, Acyclic orientations on the Sierpi´ nski gasket, available at http://arxiv.org/abs/1005.3627v1.
[10] S.-C. Chang, R. Shrock, 2003,J. Stat. Phys. 112, 815. [11] S.-C. Chang, R. Shrock,2003 ,J. Stat. Phys.112, 1019. [12] M. Rocek, R. Shrock , S,H, Tsai,1998,Physica A 252 505. [13] R. Shrock, Y. Xu,2012,J. Phys. A: Math. Theor. 45, 055212. [14] D.J.A. Welsh and C. Merino, 2000, J. Math. Phys. 41 3. [15] A. D. Sokal,2005,The multivariate Tutte polynomial (alias Potts model) for graphs and matroids,available at http://arxiv.org/abs/1005.3627v1. [16] R. Shrock,2011,J. Phys. A Math. Theor. 44, 145002. [17] J. Oxley and D.J.A. Welsh, The Tutte polynomial and percolation, in: “Graph theory and related topics”(Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), 329–339, (J.A. Bondy and U.S.R. Murty editors), Academic Press, New YorkLondon, 1979. [18] F.Jaeger, D. L.Vertigan, D. J. A.Welsh, 1990,Math. Proc.Camb. Phil. Soc. 108, 35. [19] J. Oxley,D. Welsh, 2002,Combinatorics, Probability and Computing 11, 4. [20] A. Donno, D. Iacono, The Tutte polynomial of the Sierpinski and Hanoi graphs, available at http://arxiv.org/abs/1006.5333. [21] S.-C. Chang, R. Shrock, 2001,Physica A 296, 183-233. [22] S.-C. Chang, R. Shrock, 2000,Physica A 286 ,189-238. [23] J. Salas, S.-C. Chang, R. Shrock, 2002,J. Stat. Phys. 107, 1207. [24] R. Shrock , 2000,Physica A 283,388. [25] A.-L. Barab´ asi, R. Albert 1999 Science 286, 509.
18
[26] D.J. Watts, H. Strogatz 1998 Nature (London) 393, 440. [27] S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, 2002,Phys. Rev. E 65, 066122 . [28] Z.Z.Zhang, L.L. Rong, S.G. Zhou, 2007,Physica A 377 [29] Z. Z. Zhang, S. G. Zhou, and L. C. Chen, 2007,Eur. Phys. J. B 58 337 . [30] Z.Z. Zhang, H.X. Liu, B. Wu, S.G. Zhou, 2010,Europhysics Letters 90 68002. [31] Z.Z. Zhang, Y. Qi, S.G. Zhou, W.L. Xie, J.H. Guan, 2009,Physical Review E 79 021127. [32] S.-C. Chang, L.-C. Chen and W.-S. Yang, 2007,J. Stat. Phys. 126, 649. [33] E. M. Bollt, D. ben-Avraham, 2005,New J. Phys. 7, 26 . [34] M.O. Ball, 1980,Networks 10 153 . [35] B.Mandelbrot, 1982, The Fractal Geometry of Nature Freeman ,San Francisco. [36] R.Hilfer and A.Blumen, 1984,J. Phys. A: Math. Gen. 17 L537-L545. [37] R. Albert, H. Jeong, A.-L. Barab´ asi,2000,Nature (London) 406, 378 . [38] D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, 2000,Phys. Rev. Lett. 85, 5468 . [39] R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, 2001,Phys. Rev. Lett. 86, 3682 .
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